Question 115248
Use the formula that Distance equals Speed times Time to solve this equation. In equation
form this is:
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{{{D = S*T}}}
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For Chuck, the distance he traveled was 108 miles. His speed was {{{S[c]}}} where the "c" indicates
it is Chuck's speed. Substituting these into the equation for chuck results in:
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{{{108 = S[c]*T}}}
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Solve this equation for T by dividing both sides by {{{S[c]}}} to get:
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{{{T = 108/S[c]}}}
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Meanwhile, Dana travels 99 miles in the same time. Let {{{S[d]}}} represent Dana's speed. 
Substituting these values into the distance equation results in:
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{{{99 = S[d]*T}}}
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Solve this equation for T by dividing both sides by {{{S[d]}}} and it becomes:
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{{{T = 99/S[d]}}}
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So we have two equations for T ... one for Chuck and one for Dana. But, according 
to the problem, both drivers drive for equal times. So the two T equations are equal. That
means the right sides of these equations are equal for the two drivers. So it can be said
that:
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{{{108/S[c] = 99/S[d]}}}
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Next the problem tells you that Chuck's speed is 3 miles per hour more than Dana's. So
it also can be said that:
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{{{S[c] = S[d] + 3}}}
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Substitute the right side of this into the Time equation and the result is:
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{{{108/(S[d]+3) = 99/S[d]}}}
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Put both of these fractions over a common denominator by multiplying both sides of this 
equation by {{{(S[d]*(S[d] +3))/(S[d]*(S[d] +3))}}}. Note that since the numerator of this
multiplier is equal to the denominator, this is equivalent to multiplying both sides of the
equation by 1.
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This multiplication leads to:
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{{{((S[d]*(S[d] +3))/(S[d]*(S[d] +3)))*(108/(S[d]+3)) = ((S[d]*(S[d] +3))/(S[d]*(S[d] +3)))*(99/S[d])}}}
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Cancel the like terms in the denominators and numerators on both sides:
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{{{((S[d]*(cross(S[d] +3)))/(S[d]*(S[d] +3)))*(108/cross(S[d]+3)) = (((cross(S[d]))*(S[d] +3))/(S[d]*(S[d] +3)))*(99/(cross(S[d])))}}}
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This leaves:
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{{{(108*S[d])/(S[d]*(S[d] +3))= (99*(S[d]+ 3))/(S[d]*(S[d] +3))}}}
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Since the denominators are the same on both sides, the numerators must be equal. So, setting
the numerators equal results in:
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{{{108*S[d]= 99*(S[d]+3)}}}
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Multiplying out the right side results in:
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{{{108*S[d] = 99*S[d] + 297}}}
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Subtract {{{99*S[d]}}} from both sides to get rid of that term on the right side and the
result is:
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{{{9*S[d] = 297}}}
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Solve for {{{S[d]}}} by dividing both sides by 9 to get:
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{{{S[d] = 297/9 = 33}}}
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So Dana's speed is 33 miles per hour. And Chuck's speed, which is 3 miles per hour faster,
is 36 miles per hour.
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Check. At 36 miles per hour it will take Chuck 3 hours to go 108 miles. And at 33 miles per
hour, it will take Dana 3 hours to go 99 miles. The answer checks because the times are equal.
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Hope this helps you to understand the problem and to see how it might be solved.
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